Simplify the following expression: $ q = \dfrac{-9}{10} - \dfrac{z - 3}{z + 9} $
Solution: In order to subtract expressions, they must have a common denominator. Multiply the first expression by $\dfrac{z + 9}{z + 9}$ $ \dfrac{-9}{10} \times \dfrac{z + 9}{z + 9} = \dfrac{-9z - 81}{10z + 90} $ Multiply the second expression by $\dfrac{10}{10}$ $ \dfrac{z - 3}{z + 9} \times \dfrac{10}{10} = \dfrac{10z - 30}{10z + 90} $ Therefore $ q = \dfrac{-9z - 81}{10z + 90} - \dfrac{10z - 30}{10z + 90} $ Now the expressions have the same denominator we can simply subtract the numerators: $q = \dfrac{-9z - 81 - (10z - 30) }{10z + 90} $ Distribute the negative sign: $q = \dfrac{-9z - 81 - 10z + 30}{10z + 90}$ $q = \dfrac{-19z - 51}{10z + 90}$